Key points about the cosine rule
The cosine rule is a A fact, rule, or principle that is expressed in terms of mathematical symbols. The plural of formula is formulae. used to find a missing side or angle in a triangle when two sides andAn angle between two given sides., or all the lengths of all three sides, are known.
There are two versions of the cosine rule:
- Find an unknown side using ๐ยฒ = ๐ยฒ + ๐ยฒ โ 2๐๐ cos๐ด
- Find an unknown angle using cos๐ด = \(\frac{๐ยฒ + ๐ยฒ โ ๐ยฒ}{2๐๐}\)
Scientific calculators need to be used for trigonometry and should be in degrees mode. Often there is a small D or DEG at the top of the calculator screen. If not, go into the calculator settings to change the angle units to degrees.
Make sure you are confident with finding unknown sides and angles in right-angled triangles to be successful with non-right-angled A branch of mathematics which explores the relationships between sides and angles in a triangle..
How to find an unknown side using the cosine rule
To find an unknown side in a triangle, two sides and An angle between two given sides. must be known.
Label the angles and sides of the triangle and use the formula ๐ยฒ = ๐ยฒ + ๐ยฒ โ 2๐๐ cos๐ด to find the missing side.
If the The point at which two or more lines cross. The corner of a shape. The plural form is vertices. of the triangle are not called ๐ด, ๐ต and ๐ถ, it is common practice to rename them to assist with theThe process of replacing a letter (or variable) with a number. into the formula. Make sure vertex ๐ด is opposite the side that needs to be calculated.
Answers should use the given notation in the question.
- An SAS (two sides and the included angle) triangle is a unique triangle which can be constructed with a pencil, ruler and protractor.
Follow the worked example below
GCSE exam-style questions
- Calculate the length of side ๐ฆ.
Give the answer to one decimal place.
๐ฆ = 16ยท6 cm
- Label the sides of the triangle.
Here the vertices are not labelled, so pick the vertex with angle 135ยฐ to be ๐ด. The choice of ๐ต and ๐ถ does not matter.
The 8 cm side, opposite angle ๐ถ, is called ๐.
The 10 cm side, opposite angle ๐ต, is called ๐.
The side labelled ๐ฆ, opposite angle ๐ด, is called ๐.
- Substitute the values of ๐ด, ๐, ๐ and ๐ into the formula to give
๐ฆยฒ = 10ยฒ + 8ยฒ โ (2 ร 10 ร 8)cos(135).
- 10ยฒ = 100, 8ยฒ = 64 and 2 ร 10 ร 8 = 160, so this simplifies to
๐ฆยฒ = 100 + 64 โ 160cos(135)
- Type 100 + 64 - 160cos(135) into a scientific calculator.
Usually, the calculator will automatically open a bracket after pressing the cosine button.
Remember to close the bracket after typing in the angle.
This gives ๐ฆยฒ = 277ยท1370โฆ
It is important not to round the numbers at this stage.
- The inverse of squaring is square rooting, so to find ๐ฆ, calculate the square root of 277ยท1370โฆ
Type the square root button followed by the 'ANS' button into a scientific calculator.
This gives the answer of ๐ฆ = 16ยท6474โฆ
Therefore, rounded to one decimal place, ๐ฆ = 16ยท6 cm.
- Calculate the length of ๐๐.
Give the answer to one decimal place.
๐๐ = 8ยท6 m
- Label the sides of the triangle.
Since the vertices are not called ๐ด, ๐ต and ๐ถ, let vertex ๐, with angle 67ยฐ, be ๐ด. The choice of ๐ต and ๐ถ doesn't matter.
Let vertex ๐ be ๐ต and vertex ๐ be ๐ถ.
The 9 m side, opposite angle ๐ต, is called ๐.
The 6 m side, opposite angle ๐ถ, is called ๐.
The 10 cm side, opposite angle ๐ต, is called ๐.
The side labelled ๐๐, opposite angle ๐ด is called ๐.
- Substitute the values of ๐ด, ๐, ๐ and ๐ into the formula to give
๐๐ยฒ = 9ยฒ + 6ยฒ โ (2 ร 9 ร 6)cos(67).
- 9ยฒ = 81, 6ยฒ = 36 and 2 ร 9 ร 6 = 108, so this simplifies to
๐๐ยฒ = 81 + 36 โ 108cos(67)
- Type 81 + 36 - 108cos(67) into a scientific calculator.
Usually, the calculator will automatically open a bracket after pressing the cosine button.
Remember to close the bracket after typing in the angle.
This gives ๐๐ยฒ = 74ยท8010โฆ
It is important not to round the numbers at this stage.
- Find ๐๐ by calculating the square root of 74ยท8010โฆ
Type the square root button followed by the 'ANS' button into a scientific calculator.
This gives the answer of ๐๐ = 8ยท6487โฆ
Therefore, rounded to one decimal place, ๐๐ = 8ยท6 m.
How to re-arrange the cosine formula
To use the cosine formula to find a missing angle in a triangle, the formula must be re-arranged to become:
cos๐ด = \(\frac{๐ยฒ + ๐ยฒ โ ๐ยฒ}{2๐๐} \)
Find out more about re-arranging the cosine formula below
How to find an unknown angle using the cosine rule
To find an unknown angle in a triangle, the length of all three sides must be known.
Find the missing angle by labelling the angles and sides of the triangle and using the formula:
cos๐ด = \(\frac{๐ยฒ + ๐ยฒ โ ๐ยฒ}{2๐๐} \)
This formula has vertex ๐ด as the angle to be calculated. If the variables used for the vertices are not ๐ด, ๐ต and ๐ถ, rename them to fit.
- When finding angles using trigonometry, the inverse function is used.
Follow the worked example below
GCSE exam-style questions
- Calculate the size of angle ๐.
Give the answer to one decimal place.
๐ = 102ยท6
- Label the sides of the triangle.
Since the vertices are not called ๐ด, ๐ต and ๐ถ, let vertex, ๐, the angle to be calculated, be ๐ด. The choice of ๐ต and ๐ถ doesn't matter.
Let vertex ๐ be ๐ต and vertex ๐ be ๐ถ.
The 11 m side, opposite angle ๐ด, is called ๐.
The 8 m side, opposite angle ๐ต, is called ๐.
The 6 m side, opposite angle ๐ถ, is called ๐.
Substitute the values of ๐ด, ๐, ๐ and ๐ into the rearranged formula to give cos๐ = \(\frac{8ยฒ + 6ยฒ โ 11ยฒ}{2 ร 8 ร 6} \).
Work out the value of each of the squares.
8ยฒ = 64
6ยฒ = 36
11ยฒ = 121
- Simplify the numerator and denominator.
64 + 36 โ 121 = โ 212 ร 8 ร 6 = 96
So, cos๐ = \(\frac{โ 21}{96} \).
- Work out the angle, ๐, by using the inverse function of cosine.
๐ = cosโปยน(\(\frac{โ 21}{96} \))
Press 'shift' then 'cos' to write cosโปยน on a scientific calculator.
- Type โ 21 รท 96.
Remember to close the brackets.
This gives ๐ = 102ยท6356โฆ
Rounded to 1 decimal place, Angle ๐ = 102ยท6ยฐ.
- Calculate the size of angle ๐ถ.
Give the answer to one decimal place.
๐ถ = 33ยท4ยฐ
- Label the sides of the triangle.
Although the vertices are called ๐ด, ๐ต and ๐ถ, the angle to be calculated is not ๐ด. Swap the vertices so angle ๐ถ is called ๐ด and vice-versa.
In this case, the 4ยท2 cm side, opposite angle ๐ด, is called ๐.
The 7ยท5 cm side, opposite angle ๐ต, is called ๐.
The 7 cm side, opposite angle ๐ถ, is called ๐.
- Substitute the values of ๐ด, ๐, ๐ and ๐ into the rearranged formula to give
cos๐ด = \(\frac{7ยท5ยฒ + 7ยฒ โ 4ยท2ยฒ}{2 ร 7ยท5 ร 7} \).
- Work out the value of each of the squares.
7ยท5ยฒ = 56ยท25
7ยฒ = 49
4ยท2ยฒ = 17ยท64
- Simplify the numerator and the denominator.
56ยท25 + 49 โ 17ยท64 = 87ยท61 and 2 ร 7ยท5 ร 7 = 105
So cos๐ด = \(\frac{87ยท61}{105} \).
- Work out the angle, ๐ด, using the inverse function of cosine:
๐ด = cosโปยน(\(\frac{87ยท61}{105}\))
Press 'Shift' then 'cos' to write cosโปยน on a scientific calculator.
- Type 87ยท61 รท 105.
Remember to close the brackets.
This gives ๐ด = 33ยท4486โฆ
Rounded to 1 decimal place, and remembering to swap back to the original variable, Angle ๐ถ = 33ยท4ยฐ.
Check your understanding
Quiz โ Cosine rule
Practise what you've learned about the cosine rule with this quiz.
Now you've revised the cosine rule, why not look at combined transformations and invariant points?
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